Falling Bodies

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Definition

Also known as projectile motion. It is a formula that helps to find the vertical motion of an object that is throwing straight up or down, or simply dropped under gravitational force on Earth. The formula contains the variables of height, time, acceleration (due to gravity), initial veolocity and initial height.

Worked examples

\(h(t) = -16t^2 + v_0 t + h_0\)

Standard falling-body formula (feet): height after t seconds, starting at height \(h_0\) with initial velocity \(v_0\).

\(h(t) = -4.9t^2 + 20t + 5\)

Object thrown upward at 20 m/s from 5 m high; \(-4.9\) is half of gravity in meters per second squared.

\(h(2) = -16(2)^2 + 32(2) + 10 = -64 + 64 + 10 = 10\) ft\(\)

Plug in \(t=2\) to find the height at 2 seconds.

Common mistakes

\(h(t) = -16t^2 + v_0 t + h_0\) with \(v_0 = -10\) for dropping → \(v_0 = 0\) when simply dropped (no initial velocity) Dropped means released from rest; negative \(v_0\) would mean thrown downward.
Using \(-16\) when the problem gives meters → Use \(-4.9\) for meters, \(-16\) for feet The gravity constant depends on the unit system; always match the problem's units.
\(h(t) = -16t^2 + 10\) for object thrown up at 10 ft/s → \(h(t) = -16t^2 + 10t + h_0\) Initial velocity must multiply \(t\); the constant term is initial height, not velocity.

Where you'll use it next

Falling-body formulas reappear in physics for kinematics and energy problems, calculus when modeling motion and derivatives, and real applications like optimizing projectile launch angles or designing safety systems.

Found in 1 StudyPug lesson

Position, Velocity, and Acceleration as Derivatives

Calculus 1

Discover how taking derivatives of a position function gives velocity and acceleration, and how their signs describe motion.

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